Let f(x) be a differentiable function which s | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given the functional equation $f(xy) = f(x) + f(y)$.Setting $x=1, y=1$,we get $f(1) = f(1) + f(1)$,which implies $f(1) = 0$.By the definition of t

Vedclass · Accepted Answer

$\frac{f'(1)}{x}$

Vedclass · Answer

(A) Given the functional equation $f(xy) = f(x) + f(y)$.Setting $x=1, y=1$,we get $f(1) = f(1) + f(1)$,which implies $f(1) = 0$.By the definition of the derivative:$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$Using the property $f(xy) = f(x) + f(y)$,we can write $f(x+h) = f(x(1 + \frac{h}{x})) = f(x) + f(1 + \frac{h}{x})$.Substituting this into the limit:$f'(x) = \lim_{h 	o 0} \frac{f(x) + f(1 + \frac{h}{x}) - f(x)}{h}$$f'(x) = \lim_{h 	o 0} \frac{f(1 + \frac{h}{x})}{h}$Since $f(1) = 0$,we can write $f(1 + \frac{h}{x}) = f(1 + \frac{h}{x}) - f(1)$.Let $t = \frac{h}{x}$. As $h 	o 0$,$t 	o 0$.$f'(x) = \lim_{t 	o 0} \frac{f(1+t) - f(1)}{xt} = \frac{1}{x} \lim_{t 	o 0} \frac{f(1+t) - f(1)}{t}$By the definition of the derivative at $x=1$,$\lim_{t 	o 0} \frac{f(1+t) - f(1)}{t} = f'(1)$.Therefore,$f'(x) = \frac{f'(1)}{x}$.

Let $f(x)$ be a differentiable function which satisfies the equation $f(xy) = f(x) + f(y)$ for all $x > 0, y > 0$. Then $f'(x)$ is equal to:

Similar Questions

Let $f$ be a differentiable function satisfying the relation $f(xy) = xf(y) + yf(x) - 2xy$ for all $x, y > 0$ and $f'(1) = 3$. Which of the following statements is true?

If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f(x) = 120}$,find the value of $n$.

If $f: R \rightarrow R$ is a differentiable function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$ and if $f^{\prime}(4)=24$ and $f^{\prime}(0)=3$,then $f(4)=$

The values of $b$ and $c$ for which the identity $f(x + 1) - f(x) = 8x + 3$ is satisfied,where $f(x) = bx^2 + cx + d$,are:

If $f : R \to R$ is such that $f(x + y) = f(x) + f(y)$ for all $x, y \in R$,$f(1) = 7$ and $\sum_{r=1}^{n} f(r) = 14112$,then $n$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module